Rational Suspension

: The article argues that there are different ways of justifying suspension of judgement. We suspend judgement not only privatively, that is, because we lack evidence, but also positively, that is, because there is evidence that provides reasons for suspending judgement: suspension is more than the rational fallback position in cases of insuf ﬁ cient evidence. The article applies the distinction to recent discussions about the role of suspension for inquiry, Turri ’ s puzzle about withholding, and formal representations of suspension.

WE BELIEVE A LOT, and we disbelieve a lot. With respect to most questions, however, we lack a response. Some questions we have not considered yet; others are still pending; and yet others seem not to allow for any definite answer. Most of us have no belief on whether the number of letters on this page is odd or even (we do not care), on whether we are living in a dream or not (we sometimes care, but it is difficult to say), or on whether the coin will land heads or tails (we definitely carenobody wants to wash the dishesbut we take the coin to be fair). We frequently lack a belief, and often we do so rationally.
Recent years have seen a rapidly growing interest in suspension of judgement, usually with a focus on its nature. The dominant view is that suspension of judgement is not to be characterized by mere absence but either is a sui generis indecision-representing attitude (esp. Friedman, 2013aFriedman, , 2013bFriedman, , 2017 or involves a higher-order belief (e.g., Crawford, 2004;Masny, 2020 andRaleigh, 2021). Our discussion has a different focus. We are not so much concerned with the analysis of suspension but with its rationale. The article explores different ways to justify suspension. We argue that, whereas there is but a single way in which belief or disbelief can be epistemically justified, namely by reference to positive evidence, there are different ways in which the third doxastic possibility, suspension of judgement, can be justified. More specifically, we argue that suspension is not a mere rational fallback position: besides privative suspension, which is justified by the absence of sufficient evidence (section 2), there is positive suspension, which is justified by the presence of evidence providing reasons for suspension (section 3). We will apply this distinction to recent discussions such as the debate about the role of suspension for inquiry and John Turri's "puzzle about 1 For reasons of convenience we will use the locution "S suspends judgement about p" or "S suspends judgement with respect to p," although it might be more felicitous to use the interrogative construction "S suspends judgement about whether or not p" (see Friedman, 2017, p. 304; but see Masny, 2020 for a response). 2 The first example is due to Wedgwood (2002, p. 272), the second a variation of Friedman (2013a, p. 168). 3 Friedman also tentatively suggests that the absence of belief and disbelief in p is not necessary for suspending judgement. We will not discuss this proposal here. however, suspension of judgement cannot be reduced to other (doxastic) attitudes. Instead, Friedman claims suspension of judgement is a sui generis "indecisionrepresenting" attitude: " [O]ne suspends judgment about p only if one has an attitude that expresses or represents or just is one's neutrality or indecision about which of p, not-p is true. " Friedman (2013b" Friedman ( , 2017 also talks of suspension of judgement as an "interrogative attitude" and closely relates suspension to inquiry: the content of a suspension attitude is a question (or the proposition representing the content of the question), and one suspends judgement on some matter if and only if one inquires into that matter. Different objections to Friedman's view have been put forward, for example, in Archer (2019), Masny (2020), andRaleigh (2021). Alternative first-order accounts of suspension are developed, for example, in Salmon (1986Salmon ( , 1989, Wedgwood (2002), Sturgeon (2008), Atkins (2017), and Moon (2018). 4 What unifies the metacognitive and the sui generis account is that they provide a definition of suspension in terms of the presence of a mental attitude. As McGrath (2020, p. 2) summarizes the majority view on suspension, "[to suspend judgment about p] is to be in a positive state on the question whether p, and in that sense to be an attitude." 5 We will not here discuss the two suggestions and will even be neutral on whether an attitude account of suspension is correct in the first place. However, we assume that there are preconditions for believing, disbelieving, and suspending: only those subjects can believe p, disbelieve p, or suspend judgement about p who are able to form beliefs, who can grasp and understand p, and, perhaps, who have considered p. 6 If these conditions are not fulfilled, the subject cannot adopt any position in doxastic possibility space and, as a consequence, no meaningful debate about the subject's doxastic rationality is 4 Some authors (e.g., Lord, 2020;Raleigh, 2021;Rosa, 2020) also discuss the "credal view" of suspension, which identifies suspension of judgement with a middling credence, or degree of belief; (see H ajek, 1998;Christensen, 2009). However, we think that it is essential to distinguish the following two questions: What are the coarse-grained doxastic notions? How do they relate to the fine-grained notion of credence? It is one thing to spell out what suspension is and another to suggest a bridge principle between suspension and credence. The credal view only provides a possible answer to the second question. Therefore, we disregard this account here. (See, however, our remarks on the formal representation of suspension in section 5.) 5 McGrath's own view on suspension of judgement radically differs from the majority view. He argues that suspension should not be understood as a static position in doxastic possibility space, like belief and disbelief, but as involving an agential aspect such as putting off a judgement, refraining from judging, active withholding, or inquiring (2020, pp. 4-5; for a similar view, see also Sosa, 2019). Whether it is rational to, say, put off a judgement, may depend on nonepistemic factors. Because we are exclusively concerned with questions of epistemic rationality, we consider only those accounts of suspension that understand it as a position in doxastic possibility space. 6 Note, however, that accepting "having considered a proposition" as a precondition for belief, disbelief, and suspension would prevent the possibility of implicit belief, disbelief, and suspension. possible. Thus, we accept the negative claim of the majority view: suspension of judgement is not to be identified with the mere absence of belief and disbelief. Stones do not suspend judgement, nor does the caveman with respect to the Higgs-boson proposition, because the respective subjects do not meet the preconditions for belief, disbelief, and suspension in the first place.
We will also assume, for the purpose of this article, that there are exactly three, mutually exclusive and jointly exhaustive, positions in doxastic possibility space. This results in the following definition of suspension: A subject S suspends judgement about p if and only if (i) S meets the preconditions for belief, disbelief, and suspension with respect to p; and (ii) S neither believes nor disbelieves p. We leave it open, however, whether any subject that suspends judgement, that is, any subject that fulfils the joint preconditions for belief, disbelief, and suspensionand that neither believes nor disbelieves pthereby already has some positive suspension-constituting attitude towards p. Thus, we remain neutral with respect to the positive claim of the majority view.
Our concept of suspension of judgement is meant to be little more than a terminological stipulation that provides us with a definite and unambiguous starting point for our explorations in the rationality of suspension.

Privative Suspension
We will argue that there are two fundamentally different ways in which a subject can justify suspension of judgement. Suspension can be justified either privatively or positively. Roughly put, in cases of privatively justified suspension, suspension is rational only because the subject's total evidence sufficiently supports neither p nor not-p. The subject is justified neither in believing nor in disbelieving, and only therefore indirectly justified in suspending judgement. Suspension here figures as a rational fallback position. In cases of positively, or directly, justified suspension, on the other hand, suspending judgement about p is justified by a further belief in a different proposition q: the rational subject suspends judgement not because of her lack of evidence, but because she has positive evidence providing a sufficient reason to suspend.
We begin with privative suspension. A paradigm case of privative justification for suspension is provided by balanced evidence. Here is an example of privative justification of this sort: Assume that, prior to obtaining any evidence, S is neutral with respect to whether p or not-p. Then S interviews 10 witnesses, whom she believes to be equally trustworthy and reliable. Five witnesses claim p and five claim not-p. S's evidence for and against p is in perfect equilibrium, and therefore suspension is the only rational response 7 ; S should suspend judgement about p. 8 Or assume that you have drawn 98 balls from an urn containing 100 balls. Fortynine of the drawn balls have turned out red, and the other 49 have turned out black. Do you believe that the next ball will be red or that it will be black? As your evidence is in balance, you should not take sides but suspend judgement. Suspension about p can be privatively justified in virtue of the evidence for and against p being balanced. 9 Observe also that rational suspension does not require perfect evidential balance; near equilibrium suffices. If, for example, of the 10 equally reliable and trustworthy witnesses, six assert and four deny p (or 50 drawn balls are red and 48 black), suspending judgement still seems the right, or at least a rationally permissible, response to the evidence. In cases of (nearly) balanced evidence, the total evidence favours neither p nor not-p to a sufficient degree. In such cases, suspension serves as a rational fallback position.
Another relevant factor besides the balance of evidence is the weight of the total evidence. It makes a difference whether, say, we have only interviewed three out of the 10 witnesses and one spoke in favour of p while two spoke against p, or if we have interviewed nine witnesses and three spoke in favour of p and six against p. The balance of evidence, measured by the ratio of the weights of the evidence speaking for p and the evidence speaking against p, is the same in both cases, but the total weight of evidence is not. 10 We suggest that suspension can be privatively justified in virtue of the total weight of evidence being sufficiently small. Take the extreme case and assume that S has as yet interviewed only one of the 10 witnesses, and that witness has claimed p. Although all of S's evidence speaks in favour of p, suspension still seems to be a rational doxastic response. If so, suspension again serves only as the rational fallback position: S is justified in suspending judgement only because her total evidence sufficiently supports neither p nor not-p. And analogously, if you have only drawn a single ball from the urn so far, you are justified in suspending judgement about the colour of the next ball because the weight of 7 In general, the question how individual reasons merge to determine overall justification is far from trivial. See, for example, the discussions in Lord & Maguire (2016). 8 As is common in the debate on theoretical rationality, we understand rationality norms as evaluative norms, not prescriptive ones. When we claim that you should suspend judgement about p, we do not presuppose that you can voluntarily suspend judgement. Rather, this is supposed to mean that your doxastic position is rational only if you suspend judgement about p. 9 The idea of justified suspension due to balanced evidence seems to be the driving motive of Pyrrhonian scepticism. The Pyrrhonian sceptic argues that we should suspend judgement because of isosthenia, the possibility of equally strong reasons for and against a certain proposition. 10 For a thorough discussion of the notions of balance and weight of evidence, see Joyce (2005). evidence is too low to warrant any beliefs about the colour of the ball. 11 Of course, S should have a higher degree of belief in p than in not-p after questioning only the first witness or, alternatively, drawing the first ball. But if the total weight of evidence is very low, we suggest that S is nevertheless at least rationally permitted to suspend judgement about p. 12

Positive Suspension
The epistemological debate in general, and the debate on suspension specifically, takes privatively justified suspension to be the paradigm case of suspension. Feldman and Conee (1985, p. 15), for instance, say that "suspension of judgment is the fitting attitude for each of us toward the proposition that an even number of ducks exists, since our evidence makes it equally likely that the number is odd. Neither belief nor disbelief is epistemically justified when our evidence is equally balanced." Machuca (2015, p. 177) says that "[i]t is generally thought that suspension of judgment about a proposition p is the doxastic attitude one is rationally compelled to adopt whenever the epistemic reasons for and against p are equipollent or equally credible." And Raleigh (2021, p. 2453) holds that suspension would be "the rational response to having weak or equivocal evidence." 13 Although the possibility of positive, direct reasons to suspend is not explicitly excluded, the constant focus on privative justification suggests that most authors do not regard positive suspension as important, or even entirely overlook this possibility. Comesaña (2013, p. 376) explicitly claims that "[e]vidence justifies believing or disbelieving by telling for or against the corresponding proposition, but justifies withholding by failing to tell (on balance) either for or against the corresponding proposition." Most distinctively, Schroeder (2012, p. 280) states that there "are no direct reasons to withhold." In his view, "reasons to withhold can't be evidence … because the evidence is exhausted by evidence which supports p and evidence which supports not-p. But the evidence which supports p is reason to believe p, and the evidence which supports not-p is reason to believe not-p" (Schroeder, 2012, p. 276). 14 The view that there cannot be evidence which provides a direct reason for suspension appears to be informed by the picture of an evidence scale with exactly two trays: a belief-tray and a disbelief-tray. All evidence speaking in favour of p belongs into the belief-tray, and all evidence speaking in favour of not-p belongs into the disbelief-tray. Reasons to suspend can then only be an emergent phenomenon: they result from properties of the total amount of reasons for belief and for disbelief. One such property is that the trays are filled equally (representing cases of evidential balance), and another is the property that the trays do not contain enough evidence to set the scale in motion (representing cases of insufficient evidential weight). We suggest that this picture is crucially misleading for many situations: it is not the case that all p-relevant evidence supports either p or not-p and thus belongs in either the belief-or the disbelief-tray. There can be evidence providing a direct, positive reason for suspension. To stick with the metaphor, we claim that a third tray, a suspension-tray, is required.
We will consider two cases in which a subject rationally suspends judgement about p in virtue of believing a proposition q: vagueness-driven suspension and chance-driven suspension. 15 In these cases, a q-belief positively, or directly, justifies suspending judgement about p. To allow for a straightforward description of the cases, we implicitly assume that the justifying belief, that is, the q-belief, is itself justified, and that there is no further evidence outweighing the q-belief or defeating its justificatory force.
First, vagueness-driven suspension. If S suspends judgement about a is F because she believes that a is a borderline case of F, then S is positively justified 14 The aim of Schroeder (2012) is to explore how practical factors could play a role for epistemic justification. For instance, he claims that "one important source of reasons to withhold will come from the preponderance of the cost of having a false belief over the cost of missing out on having a true belief" (Schroeder, 2012, p. 277). Although Schroeder acknowledges that such considerations do not constitute evidential reasons to suspend, he defends the view that they nevertheless constitute epistemic reasons. We will not discuss the role of non-evidential reasons here. 15 Besides cases of chance and vagueness, there might be many more scenarios giving rise to positively justified suspension. For example, one could argue that whenever S believes that p is neither true nor false, S may (or even must) rationally suspend judgement about p. Possible examples are paradoxical sentences, future contingents, or empty definite descriptions. However, we should be careful and not lump these cases together with chance and vagueness. See, for instance, Caie (2012), who argues that, if one believes p to be paradoxical, one should be in a state such that it is indeterminate whether one believes p. Another class of examples might be cases involving higher-order evidence. For discussions of the role of higher-order beliefs in justification, and the normative interrelation between firstand secondorder evidence, see, for example, Christensen (2010) and Lasonen-Aarnio (2014). See Archer (2019, section 3), for an analysis of cases in which suspension is justified by the higher-order belief that one will never be able to form a justified belief about p.
in suspending judgement. For example, fully believing that Bob is bald and fully disbelieving that Bob is bald seem both rationally incompatible with believing that Bob is borderline bald. Thus, believing that Bob is a borderline case of a bald person directly justifies suspending judgement about the proposition that Bob is bald. Suspending judgement is here not merely warranted by the fact that S's total evidence sufficiently supports neither a is F nor a is not F, although this is, of course, the case: S has no evidence either way. What drives suspension is the fact that S has positive evidence of a different sort, viz., the belief that a is a borderline case of an F. Or consider that S believes that a given ball is a borderline case of a red ball. Then S will rationally suspend judgement about the proposition that the ball is red. In these cases, suspension is not the rational reaction to lack of evidence but to the presence of evidence.
If S believes that a is a clear borderline case of an F, then suspension about whether a is F is justified. Even more, suspension seems to be rationally obligatory. But we can relax the standards for rational vagueness-driven suspension. There is not only vagueness but also higher-order vagueness. Assume that S does not believe that a is a clear borderline case of an F but that a is a borderline case of a borderline case of an F. In this case, suspension still seems to be justified. Although it might no longer be rationally obligatory to suspend judgement, it is at least rationally permissible to suspend judgement about a being F.
Let us turn to a second form of positively justified suspension, chance-driven suspension. If S's suspension of judgement about p is based on her belief that the objective chance of p is 0.5, then S is positively justified in suspending judgement. For example, S may believe that drawing balls from an urn is a genuinely chancy process, that the urn in front of her contains the same numbers of red and black balls, and thus that the chance of drawing a red ball is 0.5. What is S's rational attitude with respect to the proposition that she will draw a red ball? Obviously, S should suspend judgement about this proposition. Again, suspending judgement is not merely justified by the fact that S's total evidence sufficiently supports neither p nor not-p, although this is, of course, the case: S has no evidence either way. But, as in the vagueness case, this does not point to a doxastic lacuna. S's suspension is not due to ignorance but rather to positive evidence that generates reasons supporting suspension. S's belief that the chance of p is 0.5 provides positive justification for suspending judgement about p. Analogously, if S thinks that the chance for a radioactive atom to decay within time span t is 0.5, S will rationally suspend judgement on the proposition that the atom will decay within t. Equal-chance beliefs provide positive reasons for suspending judgement.
If S believes that the chance of p is 0.5, suspension in p is thereby justified. Even more, suspension of judgement seems rationally obligatory. But we can again relax the standard for rational suspension. The believed chance need not be 0.5 exactly. If S believes the chance of p to be, say, 0.499 or 0.502, suspension of judgement still seems justified. Although it might no longer be rationally obligatory to suspend judgement, it at least seems rationally permissible. 16 We said above that S is positively justified in suspending judgement about p if S's suspension is based on the belief that the chance of p is 0.5. S's suspension about p is epistemically based on the belief that q if and only if the belief that q is a reason why S suspends judgement about p. 17 If S suspends judgement about whether the next ball is red (p) because S believes that the chance of drawing a red ball is 0.5 (q), then S's p-suspension is based on her q-belief. Now consider the case that, at t 1 , the ball has already been drawn, and S knows this but could not get a glimpse of its colour. At t 1 , S no longer believes that the objective chance that a red ball has been drawn is 0.5; S knows that the chance for this is either 0 or 1. However, S's evidence has not changed in any relevant way between t 0 , the time before the drawing, and t 1 , the time after the drawing. S still suspends judgement about the colour of the ball at t 1 because she believes that, at t 0 , the chance of drawing a red ball was 0.5. That is, S's suspension at t 1 is still based on a chance belief, more precisely on the belief that, at t 0 , the chance of drawing a red ball was 0.5. Therefore, S's suspension is still justified positively. In order for suspension about p being based on a chance-belief, one need not believe that p still is a matter of chance. It suffices that one believes that it was a matter of chance (and one has not gained any further p-related evidence that would overrule the chance-belief). 18 Cases of chance-driven and vagueness-driven suspension show that suspension can be justified positively. Suspension is here supported by an element of the body of evidence, that is, an individual belief, and not, or not only, by the combined effect of evidence for and against p. Whereas in cases of privative suspension the subject has relevant epistemic deficitsshe does not know all the relevant factschancedriven and vagueness-driven suspension need not involve ignorance. On the contrary, S's evidential situation with respect to the propositions in question may be ideal: Assume that S is a competent speaker of English and has all the relevant information there isknows all the facts, including the objective chance of p, the exact wavelength of the reflected light, etc. 19 She may still rationally suspend judgement about p. Privative suspension is driven by ignorance, positive suspension by evidence.
Before we apply the distinction between positive and privative suspension to recent discussions in the literature, we want to address a possible objection to the distinction between privatively and positively justified suspension. The objection says that privative suspension reduces to positive suspension on the basis of the following introspection principle: If S's relevant evidence is balanced/has a low total weight, then S believes that her relevant evidence is balanced/has a low total weight. If this principle holds, then even in alleged cases of merely privatively justified suspension, for example, cases of balanced evidence or of insufficient total weight, there is a belief that directly justifies suspending judgement, namely the meta-belief that the relevant firstorder evidence is balanced/has a low total weight. This would suggest that privatively justified suspension reduces to positively justified suspension because there is always a belief that directly warrants suspension. There are, according to this objection, no cases of genuine privative justification for suspension.
We have a two-part reply: First, the objection is based on a highly contested introspection principle. It seems plausible that even a perfectly rational agent need not necessarily have a belief about evidential weight or comparative evidential strengths. S's evidence for p and for not-p can be balanced without S having the belief that her evidence is balanced. Second, for those accepting the introspection principle, we can slightly reformulate the characterization of positively justified suspension and thus protect it from the current objection. According to this reformulation, suspending judgement is positively justified if and only if there is a suspension-supporting firstorder belief. Because the belief that one's own evidence is balanced/has a low weight is a second-order belief, there is no positive suspension in this redefined sense, even if the subject's justification for suspending judgement is this second-order belief.

Privative Versus Positive Suspension: Applications
In this section, we will put the distinction between positive and privative suspension to work by applying it to two recent discussions in the literature: first, the debate about the relation between suspension and inquiry; and second, Turri's (2012) so-called puzzle about withholding.

Suspension and inquiry
Many authors think there is an intimate relation between suspension and inquiry, although there is fundamental disagreement about how they are related. 20 20 We find a discussion of the connection between suspension and inquiry already in Sextus Empiricus, the main recorder of Pyrrhonian scepticism (see Annas and Barnes, 1994). For recent discussion about the relation between suspension and inquiry, see, for example, Friedman (2013aFriedman ( , 2013bFriedman ( , 2017Friedman ( , 2019, Wieland (2014), Machuca (2015), Archer (2019), Sosa (2019), and McGrath (2020).
According to one view, one suspends judgement about p if and only if one inquires into whether p. Suspending judgement about whether p amounts to being in an inquiring state of mind with respect to whether p, that is, roughly put, having the question on one's research agenda, being sensitive to relevant evidence, etc. 21 According to another view, suspension of judgement is understood as one of three possible ends of an inquiry: we can end an inquiry about p by coming to believe p, coming to disbelieve p, or by suspending judgement about p. Suspension is here seen as an inquiry-stopping attitude. 22 Given our distinction, the response is straightforward. Positive suspension can serve as one way of ending an inquiry, whereas privative suspension usually fosters inquiry. For illustration, consider Ann, who wants to know whether Bob, the nice guy from the chatroom, is bald or not. She skips through Bob's pictures. Unfortunately, he is wearing a cap on most of them. But then, there he is, holding his head right into the camera. Ann finally has a perfect view of Bob's present hair situation. But she still does not know what to feel or think: Bob is a borderline case of a bald person. Ann continues suspending with respect to whether Bob is bald. But she will stop skipping through his pictures (at least for this purpose). Before Ann looked at Bob's pictures, she had been privatively justified in suspending belief about whether he is bald or not. She had no evidence either way. After she found the relevant picture, she is positively justified in suspending. Her suspension is no longer merely due to ignorance but to her vagueness-related belief. Therefore, she will not continue her inquiring activities, and she will no longer be in an inquiring state of mind.
If S fully believes that a is a borderline case of an F, or that the chance of a's being F is 0.5, there is often no reason for S to further inquire into whether p. Positive suspension can, and often does, serve as one way of ending an inquiry. 23 On the other hand, a subject who privatively suspends judgement about p will continue her inquiry as long she wants to decide p. She only suspends because of her lack of sufficient reasons to believe or disbelieve p. Privative suspension is 21 Friedman claims that "one is inquiring into some matter if and only if one is suspended on the matter" (2017, p. 302) and that "suspension of judgement can itself be thought of as an inquiring attitude" (2017, p. 320; see also Friedman, 2019). Wieland (2014, p. 234) argues that "it is rational to suspend one's belief regarding p so long as one is open to investigating p." 22 Along these lines, Bertrand Russell proposed that "[suspension of judgement] represents the result of an attempt to decide between belief and disbelief" (Russell, 1913, p. 143). Somewhat surprisingly, we find similar remarks in Friedman's earlier paper (2013a, p. 179): "So a subject can move from a state of deliberating about p to a more settled state of agnosticism by suspending judgement about p. Suspending judgement then can be thought of as one way of terminating a deliberative process and (other things equal) moving into a more settled state." 23 We do not claim that a positive justification for suspension always serves as an inquiry-stopper. If one suspends judgement about whether the coin landed heads, based on one's belief that the chance of heads, prior to the toss, was 0.5, one might still be in an inquiring state of mind with respect to the outcome of the coin toss. but a transitory state, at least ideally, one which gives way to belief, disbelief, or positive suspension as soon as one attains sufficient evidence. Once the distinction between privative and positive suspension is made explicit, different positions regarding the relation between suspension and inquiry naturally fall into place.

Turri's puzzle
The distinction between positive and privative suspension shows that there is a structural difference between justifying belief or disbelief and justifying suspension. All three doxastic possibilities, viz., belief, disbelief, and suspension, can be justified positively; however, at least with respect to most propositions only suspension can be justified privatively. There cannot be privative justification of belief or disbelief: S cannot be justified in believing p solely on the basis of lacking sufficient justification for both disbelief and suspension. Privative justification is peculiar to suspension of judgement. Suspension can be justified without there being decisive positive evidence providing reasons for it, but belief or disbelief cannot. 24 Or, to phrase it somewhat differently: in cases of positive justification of suspension, one is unjustified both in believing and in disbelieving because one is justified in suspending; in cases of privative justification of suspension, one is, conversely, justified in suspending because one is unjustified both in believing and in disbelieving. However, one is never justified in believing (or in disbelieving, respectively) merely because one is unjustified both in disbelieving and suspending (or in believing and suspending, respectively). There is no privative justification for belief or for disbelief. 25 This also explains Turri's (2012) "puzzle of withholding." Turri describes the following case: The best mathematicians gather in a secret meeting and try to prove or disprove a mathematical statement p. When they leave the room, they all tell you not to suspend judgement about p. You trust them and have no additional evidence as to whether p. What should you do? Should you believe, disbelieve, or suspend judgement about p? Turri (2012, p. 359) argues as follows: "If all the Mathematicians had said that believing is not the thing to do, then it would have been true that you shouldn't believe. And had all the 24 This does perhaps not hold for all propositions. Existence propositions such as "God exists" or "There is a china teapot revolving around the sun" (Russell, 1952) might be good candidates for propositions for which, in the absence of any evidence, disbelief is the rational fallback position. Some philosophers might perhaps also argue that there are some existential propositions, such as "The world exists" or, again, "God exists," for which belief is the rational fallback position. With respect to ordinary propositions, however, it is widely agreed that, in the absence of evidence, suspension is the correct doxastic response. 25 We are grateful to an anonymous referee of this journal for putting it this way.
Mathematicians said that disbelieving is not the thing to do, then it would have been true that you shouldn't disbelieve. It stands to reason, then, that if all the Mathematicians say that withholding is not the thing to do, then you shouldn't withhold. This is puzzling." The puzzle arises from an assumed symmetry in the justification conditions of belief, disbelief, and suspension. But, as we have argued, there is a relevant asymmetry in the rationality conditions of belief / disbelief on the one hand, and suspension of judgment on the other: only suspension, but not belief or disbelief, can be justified privatively. In each of the three scenarios, that is, whether the mathematicians tell you not to believe, not to disbelieve, or not to suspend, you lack positive reasons for either belief, disbelief, or suspension. You are neither positively justified to believe, nor to disbelieve, nor to suspend judgement. There is no asymmetry with respect to positive justification. However, in all three cases you lack sufficient reason to either believe or disbelieve. Now, whenever you lack sufficient reason to believe or disbelieve, you are privatively justified to suspend judgement. Therefore, you should suspend judgement in all three cases. In particular, you should suspend despite the fact that the mathematicians tell you not to. Granted, you lack positive justification for suspension, but privative justification suffices. As said above, this is not the case for belief. One is never justified in believing only because one is not justified in suspending and disbelieving (and analogously for disbelieving). When it comes to privative justification, on the one hand there is a relevant asymmetry between believing and disbelieving and on the other a suspension. Turri's puzzle is resolved by the fact that only suspension allows for privative justification. 26

Outlook: Suspension and Degrees of Belief
Let us briefly review. We distinguished mere absence of both belief and disbelief from suspension of judgement (section 1), and we distinguished, within the category of suspension of judgement, privatively justified suspension (section 2) from positively justified suspension (section 3). Finally, the distinction between privative and positive suspension was applied to recent discussions about inquiry and to Turri's puzzle (section 4).
We think that the above distinctions might also have ramifications for the formal representation of suspension of judgement in terms of degrees of belief. We will therefore conclude with a tentative outlook on the relation between the categorical notion of suspension of judgement and the quantitative notion of degrees of belief. 27 Consider cases of mere non-belief first, that is, cases in which the subject does not meet the preconditions for believing p, disbelieving p, and suspending judgement about p. In such cases, the subject either lacks the conceptual resources to grasp p or maybe just has not yet considered p. The mere absence of belief, in this sense, is arational. We think that this fact is best captured by assigning no degree of belief to p at all. If p is not among the propositions that S can or actually does grasp, S takes no stand with respect to p, and S's degrees-of-belief function should therefore not assign any degree of belief to p. We suggest that only gappy degrees-of-belief functions can adequately model mere absence of belief. 28 Next, consider the more interesting case of positively justified suspension. We discussed two notions of positively justified suspension: vagueness-driven suspension and chance-driven suspension. We hold that vagueness-driven and chance-driven suspension differ in their compositional behaviour. For instance, two rational subjects who individually suspend judgement about a is F and a is G will have different doxastic attitudes towards the conjunctive proposition a is F & a is G, depending on whether their suspension in a is F and a is G is chance-driven or vagueness-driven. Therefore, we will suggest, chance-driven and vagueness-driven suspension demand different formal representations. A simple example will illustrate this.
Both Charlie and Viola individually suspend judgement about a is F and a is G. Charlie's suspension about both propositions is chance-driven: he believes that a has a chance of 0.5 to be F and the same chance to be G. Viola suspends judgement about both propositions, too, but her suspension is vagueness-driven: she believes (with full certainty) that a is a borderline case of an F and that a is a borderline case of a G. What are Viola's and Charlie's rational doxastic positions with respect to the conjunctive proposition a is F & a is G? Because Charlie believes that the chance of each conjunct is 0.5, he should believe that the conjunctive proposition has a chance of 0.25 (we may assume that he knows the conjuncts to be probabilistically independent). Charlie is not fully neutral with respect to the conjunctive proposition a is F & a is G but rather tends to disbelieve it. How about Viola? As we described the case, Viola is fully certain that a is both a borderline case of an F and a borderline case of a G. She therefore tends to believe neither that the conjunction 27 For an alternative view on the logic of suspension, see Rosa (2019Rosa ( , 2021. 28 See Decker (2012) for a related suggestion. is true nor that it is false. Viola should thus assign a degree of belief of 0.5 also to the conjunctive proposition. 29 Vagueness-driven and chance-driven suspension differ in their compositional behaviour. In principle, there seem to be two ways to account for this: One could insist on modelling chance-driven and vagueness-driven suspension within a single formal system, for example, subjective probability theory, but reject the idea that there is a unique bridge principle between degrees of belief and the categorical notion of suspension of judgement. However, according to this proposal, Viola and Charlie would have the same degrees of belief in the conjunctive proposition, which, as we have argued above, seems to be the wrong result. We therefore suggest another route. The different ways to justify suspension seem to point to different dimensions along which belief can be graded: there are chance-driven and vagueness-driven degrees of belief. 30 To model chance-driven suspension, we need the former, to model vagueness-driven suspension, the latter. Although full-blown suspension can perhaps be modelled by a degree of belief of 0.5 irrespectively of whether we have a case of chance-driven or vagueness-driven suspension, we should be careful not to lump these cases together into one formal model. To model Charlie's doxastic state, we need chance-driven degrees of belief, which would follow probability theory, whereas to model Viola's doxastic state we have to employ vagueness-driven degrees of belief, which would have to follow a logic of vagueness.
The logics of our two examples of positively justified suspension follow from the contents of the justifying beliefs: probability theory for chance-driven suspension and a logic of vagueness for vagueness-driven suspension. But what is the correct logic for privatively justified suspension? In cases of privatively justified suspension, there are no suspension-justifying beliefs; rather, one suspends only because one lacks reasons to believe or disbelieve. Here, suspending judgement is due to genuine epistemic uncertainty. Whether the logic of genuine epistemic uncertainty is best captured by classical probability theory or, say, by ranking theory, 31 is not the topic of this article.
We therefore conclude with the suggestion that there cannot be a unique formal representation of suspension of judgement: the different ways to suspend judgement must be modelled with different kinds of degrees of belief. We should not look for the logic of suspension but explore the many types of suspension, their various formal renderings, and the different associated ways to grade belief. 29 See Schiffer (1998, pp. 204-205), Schiffer (2000, pp. 224-227), and Wright (2006) for a more detailed defence. 30 That there are degrees of belief related to vagueness is also defended in Schiffer (2000) and Smith (2010Smith ( , 2014. 31 See Spohn (2012).